Brazilian Journal of Probability and Statistics

Inventory model of type $(s,S)$ under heavy tailed demand with infinite variance

Aslı Bektaş Kamışlık, Tülay Kesemen, and Tahir Khaniyev

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this study, a stochastic process $X(t)$, which describes an inventory model of type $(s,S)$ is considered in the presence of heavy tailed demands with infinite variance. The aim of this study is observing the impact of regularly varying demand distributions with infinite variance on the stochastic process $X(t)$. The main motivation of this work is, the publication by Geluk [Proceedings of the American Mathematical Society 125 (1997) 3407–3413] where he provided a special asymptotic expansion for renewal function generated by regularly varying random variables. Two term asymptotic expansion for the ergodic distribution function of the process $X(t)$ is obtained based on the main results proposed by Geluk [Proceedings of the American Mathematical Society 125 (1997) 3407–3413]. Finally, weak convergence theorem for the ergodic distribution of this process is proved by using Karamata theory.

Article information

Braz. J. Probab. Stat., Volume 33, Number 1 (2019), 39-56.

Received: September 2016
Accepted: September 2017
First available in Project Euclid: 14 January 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Semi-Markovian inventory model of type $(s,S)$ heavy tailed distributions with infinite variance regular variation renewal reward process asymptotic expansion Karamata theorem


Bektaş Kamışlık, Aslı; Kesemen, Tülay; Khaniyev, Tahir. Inventory model of type $(s,S)$ under heavy tailed demand with infinite variance. Braz. J. Probab. Stat. 33 (2019), no. 1, 39--56. doi:10.1214/17-BJPS376.

Export citation


  • Aliyev, R. T. (2016). On a stochastic process with a heavy tailed distributed component describing inventory model of type $(s,S)$. Communications in Statistics—Theory and Methods 46, 2571–2579.
  • Aliyev, R. T. and Khaniyev, T. (2014). Asymptotic expansions for the moments of a semi Markovian random walk with Gamma distributed interference of chance. Communications in Statistics—Theory and Methods 39, 130–143.
  • Asmussen, S. (2000). Ruin Probabilities. Singapore: World Scientific Publishing.
  • Bimpikis, K. and Markasis, M. G. (2015). Inventory pooling under heavy tailed demand. Management Science 62, 1800–1813.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge: Cambridge University Press.
  • Borokov, A. A. and Borokov, K. A. (2008). Asymptotic Analysis of Random Walks, Heavy Tailed Distributions. New York: Cambridge University Press.
  • Brown, M. and Solomon, H. A. (1975). Second order approximation for the variance of a renewal reward process and their applications. Stochastic Processes and their Applications 3, 301–314.
  • Chen, F. and Zheng, Y. S. (1997). Sensitivity analysis of an inventory model of type $(s,S)$ inventory model. Operation Research and Letters 21, 19–23.
  • Chevalier, J. and Austan, G. (2003). Measuring prices and price competition online: and Quantitative Marketing and Economics 1, 203–222.
  • Embrechts, P., Kluppelberg, C. and Mikosh, T. (1997). Modeling Extremal Events for Insurance and Finance. Berlin: Springer.
  • Feller, W. (1971). Introduction to Probability Theory and Its Applications II. New York: John Wiley.
  • Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy Tailed and Subexponential Distributions. Series in Operations Research and Financial Engineering. New York: Springer.
  • Gaffeo, E., Antonello, E. S. and Laura, V. (2008). Demand distribution dynamics in creative industries: The market for books in Italy. Information Economics and Policy 20, 257–268.
  • Geluk, J. L. (1997). A renewal theorem in the finite-mean case. Proceedings of the American Mathematical Society 125, 3407–3413.
  • Gikhman, I. I. and Skorohod, A. V. (1975). Theory of Stochastic Processes II. Berlin: Springer.
  • Kesemen, T., Aliyev, R. and Khaniyev, T. (2013). Limit distribution for semi Markovian random walk with Weibull distributed interference of chance. Journal of Inequalities and Applications 133, 1–8.
  • Khaniyev, T. and Aksop, C. (2013). Asymptotic results for an inventory model of type $(s,S)$ with generalized beta interference of chance. TWMS Journal of Applied and Engineering Mathematics 2, 223–236.
  • Khaniyev, T. and Atalay, K. D. (2010). On the weak convergence of the ergodic distribution for an inventory model of type $(s,S)$. Hacettepe Journal of Mathematics and Statistics 39, 599–611.
  • Khaniyev, T., Kokangul, A. and Aliyev, R. (2013). An asymptotic approach for a semi Markovian inventory model of type $(s,S)$. Applied Stochastic Models in Business and Industry 29, 439–453.
  • Resnick, S. I. (2006). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Series in Operations Research and Financial Engineering. New York: Springer.
  • Sahin, I. (1983). On the continuous-review $(s,S)$ inventory model under compound renewal demand and random lead times. Journal of Applied Probability 20, 213–219.
  • Seneta, E. (1976). Regularly Varying Functions. New York: Springer.
  • Smith, W. L. (1959). On the cumulants of renewal process. Biometrika 46, 1–29.